# Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

## TL;DR

This paper investigates asymmetric large deviations in empirical observables derived from independent random variables, revealing how different scalings occur for rare events above or below typical values, with insights from Sanov's theorem and renormalization.

## Contribution

It unifies the analysis of asymmetric large deviations for various empirical observables using Sanov's theorem and explores their physical interpretation through renormalization.

## Key findings

- Asymmetric large deviations occur for empirical maxima, averages, and moments.
- Sanov's theorem provides a unifying framework for analyzing these deviations.
- The physical meaning of rate functions is discussed via renormalization.

## Abstract

In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we revisit the simpler observables based on independent random variables, namely the empirical maximum, the empirical average, the empirical non-integer moments or other additive empirical observables, in order to describe the cases where asymmetric large deviations already occur. The unifying starting point to analyze the large deviations of these various empirical observables is given by the Sanov theorem for the large deviations of the empirical histogram : the rate function corresponds to the relative entropy with respect to the true probability distribution and it can be optimized in the presence of the appropriate constraints. Finally, the physical meaning of large deviations rate functions is discussed from the renormalization perspective.